Decoherent histories quantum theory is reformulated with the assumption that there is one ``real'' fine-grained history, specified in a preferred complete set of sum-over-histories variables. This real history is described by embedding it in an ensemble of comparable imagined fine-grained histories, not unlike the familiar ensemble of statistical mechanics. These histories are assigned extended probabilities, which can sometimes be negative or greater than one. As we will show, this construction implies that the real history is not completely accessible to experimental or other observational discovery. However, sufficiently and appropriately coarse-grained sets of alternative histories have standard probabilities providing information about the real fine-grained history that can be compared with observation. We recover the probabilities of decoherent histories quantum mechanics for sets of histories that are recorded and therefore decohere. Quantum mechanics can be viewed as a classical stochastic theory of histories with extended probabilities and a well-defined notion of reality common to all decoherent sets of alternative coarse-grained histories.
Entangled history theory is based on the framework of consistent histories theory, which is considered to be a way to reformulate quantum mechanics without the postulation of collapsing of states. Since it was proposed in 2015, the entangled history was verified experimentally with
3 time nodes GHZ-like entangled histories in 2016. Here we extend previous results of entangled histories to arbitrary number of time nodes. For even number time nodes, we define a Bell-type inequality to distinguish quantum entangled histories and the classical histories and prove the inequality for arbitrary even time nodes histories. For odd time nodes cases, we propose a GHZ type experiment and define a function to distinguish quantum entangled and classical histories. We prove a minimal formula for the function of classical histories, which is larger than the function outcome of entangled histories.